(*^
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Setup:
:[font = input; preserveAspect; endGroup]
Needs["Algebra`Trigonometry`"]
Needs["Calculus`FourierTransform`"]
Needs["Graphics`Graphics`"]
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Problem 1: Fourier Transforms by trial and error
:[font = text; inactive; preserveAspect]
The goal of this exercise is to give the students a feeling for the Fourier transform by giving them the task of constructing a specified "goal" function using a series of sine waves.  
:[font = subsection; inactive; preserveAspect; cellOutline; startGroup]
Setup:
:[font = input; preserveAspect]
Clear["Global`*"];
:[font = text; inactive; preserveAspect]
This just defines some simple functions so that the repetitive task of comparing our sine series with the goal function is simplified. 
:[font = input; preserveAspect]
fun[c0_,c1_,c2_,c3_]:= 
 ( c0 
 + c1 Sin[ Pi 1 x]
 + c2 Sin[ Pi 2 x]
 + c3 Sin[ Pi 3 x]
 )
:[font = input; preserveAspect; endGroup]
doit[c0_,c1_,c2_,c3_]:= 
  Plot[ {goal[x],fun[c0,c1,c2,c3]} //Evaluate 
       ,{x,-1,1}
       ,PlotStyle->{RGBColor[1,0,0],RGBColor[0,0,1]}
       ];
:[font = subsection; inactive; preserveAspect; cellOutline; startGroup]
Part a)  goal[x]= Ceiling[x]-1/2
;[s]
2:0,0;18,1;33,-1;
2:1,16,12,Times,1,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = text; inactive; preserveAspect]
Here is our goal function. 
:[font = input; preserveAspect]
Clear[goal];
goal[x_]=Ceiling[x *0.99]-1/2

Plot[goal[x] ,{x,-1,1}];
:[font = text; inactive; preserveAspect]
Here I plot out the different terms in our series, one at a time, and compare this with the goal function. 
:[font = input; preserveAspect]
doit[1,0,0,0];
:[font = input; preserveAspect]
doit[0,1,0,0];
:[font = input; preserveAspect]
doit[0,0,1,0];
:[font = input; preserveAspect]
doit[0,0,0,1];
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
Your challenge: try and reproduce the goal function as closely as possible by hand. Then have a look at the "answer" given by  NFourierTrigSeries.
;[s]
4:0,0;127,1;145,0;146,1;147,-1;
2:2,16,12,Times,0,14,65535,0,0;2,13,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect]
doit[0,0.5,0.5,0.5];
:[font = text; inactive; preserveAspect]
Here is the answer. (I'll hide this.)
:[font = input; inactive; preserveAspect; endGroup]
NFourierTrigSeries[goal[x],{x,-1,1},3] //Chop
:[font = subsection; inactive; preserveAspect; cellOutline; startGroup]
Part b)  goal[x]= x 
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
Your challenge: try and reproduce the goal function as closely as possible by hand. Then have a look at the "answer" given by  NFourierTrigSeries.
;[s]
4:0,0;127,1;145,0;146,1;147,-1;
2:2,16,12,Times,0,14,65535,0,0;2,13,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect]
Clear[goal];
goal[x_]=x

Plot[goal[x] ,{x,-1,1}];
:[font = input; preserveAspect]
doit[0,0.5,0.5,0.5];
:[font = input; inactive; preserveAspect; endGroup]
NFourierTrigSeries[goal[x],{x,-1,1},3] //Chop
:[font = subsection; inactive; preserveAspect; cellOutline; startGroup]
Part c)  goal[x]= 1-Abs[x]
;[s]
2:0,0;18,1;27,-1;
2:1,16,12,Times,1,14,0,0,0;1,13,10,Courier,1,12,0,0,0;
:[font = text; inactive; preserveAspect; fontColorRed = 65535]
Your challenge: try and reproduce the goal function as closely as possible by hand. Then have a look at the "answer" given by  NFourierTrigSeries.
;[s]
4:0,0;127,1;145,0;146,1;147,-1;
2:2,16,12,Times,0,14,65535,0,0;2,13,10,Courier,1,12,65535,0,0;
:[font = input; preserveAspect]
Clear[goal];
goal[x_]=1-Abs[x]

Plot[goal[x] ,{x,-1,1}];
:[font = input; preserveAspect; endGroup]
doit[0,0.5,0.5,0.5];
:[font = input; inactive; preserveAspect; endGroup]
NFourierTrigSeries[goal[x],{x,-1,1},3] //Chop
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 2: Fourier expansion of a saw-tooth
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = subsection; inactive; preserveAspect]
Part a) 
:[font = text; inactive; preserveAspect]
Here is our goal function: f[x].
:[font = input; preserveAspect]
Clear[f];
f[x_]:=x /; x <= N[Pi];
f[x_]:=0 /; x >= N[Pi];
:[font = input; preserveAspect]
Plot[f[x],{x,0,2 Pi}];
:[font = text; inactive; preserveAspect]
We'll do the first coefficient by hand since this is a special case. 
:[font = input; preserveAspect]
c[0]= 1/(2 Pi) Integrate[ x  ,{x,0,1 Pi}]
:[font = text; inactive; preserveAspect]
The rest of the coefficients are given by:
:[font = input; preserveAspect]
c[m_]= 1/(2 Pi) Integrate[ x Exp[- I m x] ,{x,0,1 Pi}]
:[font = text; inactive; preserveAspect]
Here's a rule to help Mathematica  simplify our expressions. 
;[s]
3:0,0;22,1;33,0;62,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = input; preserveAspect]
rule={Exp[+2 I Pi m] ->  1
     ,Exp[-2 I Pi m] ->  1
     ,Exp[+  I Pi m] -> (-1)^m
     ,Exp[-  I Pi m] -> (-1)^m
     };
:[font = text; inactive; preserveAspect]
Given the coefficients, we can now create our series. 
:[font = input; preserveAspect]
(* Sometimes we have to work out special cases 
 to avoid divide by zero and other problems. *) 
 
term[0]= c[0];
:[font = input; preserveAspect]
term[1]= Limit[ c[m] Exp[I m x] + c[-m]  Exp[-I m x] ,m->1]
:[font = input; preserveAspect]
term[m_]= c[m] Exp[I m x] + c[-m]  Exp[-I m x] //.rule //Together
:[font = text; inactive; preserveAspect]
Sum the series. 
:[font = input; preserveAspect]
series[n_]:= Sum[ term[m] ,{m,0,n}]
:[font = text; inactive; preserveAspect]
and plot it for a range of terms. 
:[font = input; preserveAspect]
Plot[ Join[{f[x]},Table[series[i],{i,2,5,1}] ] //Evaluate
  ,{x,0,2 Pi}];
:[font = text; inactive; preserveAspect]
Even for 30 terms, we have trouble describing the sharp features of our function. 
:[font = input; preserveAspect]
Plot[ {f[x], series[30] } //Evaluate ,{x,0,2 Pi}];
:[font = text; inactive; preserveAspect]
While we have taken this to be an exponential series, it is trivial to convert this into a Trig series. 
:[font = input; preserveAspect]
term2[0] = term[0] //ComplexToTrig //Simplify;
term2[1] = term[1] //ComplexToTrig //Simplify;
:[font = input; preserveAspect]
term2[m_]= term[m] //ComplexToTrig //Simplify
:[font = input; preserveAspect]
series2[n_]:= Sum[ term2[m] ,{m,0,n}]
:[font = input; preserveAspect]
series[5]
:[font = input; preserveAspect; endGroup]
series2[5]
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981]
Problem 3: Repeat above, with the function f[x]=x over x=[0, 2 Pi].
:[font = section; inactive; Cclosed; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981; startGroup]
Example 4:  Relate continous and discrete Fourier Transform
:[font = text; inactive; preserveAspect]
This is a simple exercise to help student understand the extension of a discrete Fourier transform to a continous one.
:[font = input; preserveAspect]
Clear["Global`*"]
:[font = text; inactive; preserveAspect]
Pick some Fourier coefficients:
:[font = input; preserveAspect]
c={5,0,2,0,3};
BarChart[c,PlotRange->{{0, 10},All}];
:[font = text; inactive; preserveAspect]
Make a Fourier Series:
:[font = input; preserveAspect]
Clear[f];
f[x_]= Sum[ c[[n]] Sin[n x] ,{n,1,5}]
:[font = text; inactive; preserveAspect]
Make a data table from this function. 
:[font = input; preserveAspect]
data= Table[f[x] //N,{x,0,2 Pi,2 Pi/2^7}];
data //Short
:[font = text; inactive; preserveAspect]
Plot the Fourier Series:
:[font = input; preserveAspect]
ListPlot[data ,PlotJoined->True];
:[font = text; inactive; preserveAspect]
Fourier Transform Series:
:[font = input; preserveAspect]
fdata=Fourier[data] //Abs //Chop;

fdata //Short
:[font = input; preserveAspect]
(* 

Note, I Dropped the c_0 element so that the 
numbers match up better; i.e., c_1 is at 1, ... 
Compare this with first plot.

*)

BarChart[fdata //Drop[#,1]& 
     ,PlotRange->{{0, 10},All}
     ];
:[font = text; inactive; preserveAspect]
Compare with:
:[font = input; preserveAspect]
BarChart[c,PlotRange->{{0, 10},All}];
:[font = text; inactive; preserveAspect]
Same thing with ListPlot instead. 
(ListPlot is more general for our purposes.)
;[s]
3:0,0;35,1;79,0;80,-1;
2:2,16,12,Times,0,14,0,0,0;1,16,12,Times,2,14,0,0,0;
:[font = input; preserveAspect]
ListPlot[fdata //Drop[#,1]& 
     ,PlotJoined->True
     ,PlotRange->{{0,10},All}
     ,AxesOrigin->{0,0}
     ];
:[font = input; preserveAspect; endGroup]
LinearLogListPlot[fdata //Drop[#,1]& 
     ,PlotJoined->True
     ,PlotRange->All
     ];
:[font = section; inactive; preserveAspect; cellOutline; backColorRed = 58981; backColorGreen = 58981; backColorBlue = 58981]
Problem 5:  Repeat above, with  c= Table[Exp[-n^2/5],{n,1,5}]
^*)